What is Hyperconjugation ?

Hyperconjugation is the stabilising interaction that results from the interaction of the electrons in a s-bond (usually C-H or C-C) with an adjacent empty (or partially filled) p-orbital or a p-orbital to give an extended molecular orbital that increases the stability of the system. Based on the valence bond model of bonding, hyperconjugation can be described as "double bond - no bond resonance" but it is not what we would "normally" call resonance, though the similarity is shown below. What is the key difference between hyperconjugation and resonance ? Hyperconjugation is a factor in explaining why increasing the number of alkyl substituents on a carbocation or radical centre leads to an increase in stability.

Let's consider how a methyl group is involved in hyperconjugation with a carbocation centre.
First we need to draw it to show the C-H s-bonds. Note that the empty p orbital associated with the positive charge at the carbocation centre is in the same plane (i.e. coplanar) with one of the C-H s-bonds (shown in blue.)
This geometry means the electrons in the s-bond can be stabilised by an interaction with the empty p-orbital of the carbocation centre. (this diagram shows the similarity with resonance and the structure on the right has the "double bond - no bond" character)

The stabilisation arises because the orbital interaction leads to the electrons being in a lower energy orbital

Of course, the C-C s-bond is free to rotate, and as it does so, each of the C-H s-bonds in turn undergoes the stabilising interaction.
So the ethyl cation has 3 C-H s-bonds that can be involved in hyperconjugation.
The more hyperconjuagtion there is, the greater the stabilisation of the system.
So for example, the t-butyl cation has 9 C-H s-bonds that can be involved in hyperconjugation. Hence (CH₃)₃C+ is more stable than CH₃CH₂+
The effect is not limited to C-H s-bonds, appropriate C-C s-bonds can also be involved in hyperconjugation.